Radiation from Transmission Lines PART I: Free Space
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1 Radiation from transmission lines PART I: free space transmission lines Reuven Ianconescu∗, Vladimir Vulfin† ∗Shenkar College of Engineering and Design, Ramat Gan, Israel, [email protected] †Ben-Gurion University of the Negev, Beer Sheva 84105, Israel, [email protected] Abstract—This work derives exact expressions for the radiation radiation from open ended twin lead TL in free space, at the from two conductors non isolated TEM transmission lines of resonance frequencies is calculated. any cross section in free space. We cover the cases of infinite, A more conclusive and full analysis is presented in the 1951 semi-infinite and finite transmission lines and show that while an infinite transmission line does not radiate, there is a smooth paper “Radiation Resistance of a Two-Wire Line” [4]. This transition between the radiation from a finite to a semi-infinite work calculates the radiation resistance of a twin lead loaded transmission line. Our analysis is in the frequency domain and we at its termination by any impedance, considering TL ohmic consider transmission lines carrying any combination of forward losses as well. In this work we consider 0 ohmic losses, but and backward waves. The analytic results are validated by suc- generalize [4] as follows: cessful comparison with ANSYS commercial software simulation results, and successful comparisons with other published results. • We consider the separate radiation of the forward and backward waves and also their combination. From such analysis it comes out that the interference term between Index Terms—electromagnetic theory, guided waves, TEM waveguides, radiation losses, EM Field Theory. the waves does not contribute to the radiated power. • We analyze the radiation from semi-infinite TL, and show the connection with the finite TL case. I. INTRODUCTION • We do not limit ourselves to the twin lead cross section, The aim of this work is to calculate the radiated power and present a general algorithm for TL of any cross from two conductors transmission lines (TL) in free space. section. We consider any cross section of small electric size (shown in • We develop a more accurate radiation resistance. Figure 1) and TL of any length, analyzing the infinite, semi- Additional works on the subject can be found in [5]–[7]. It infinite and finite TL cases. Part II is the generalization of this is interesting to remark that [8] claims that balanced TL do work for TL inside dielectric insulator and will be published not radiate. Although [8] is only an educational document of separately. Some preliminary results of this work have been a university, such inaccuracy is a symptom showing that the presented in [1], [2]. subject of radiation from TL is not well enough known in the electromagnetic community, requiring additional research on this subject. There are two appendices in this work. In Appendix A we calculate the far potential vector from a general cross section TL (see Figure 1), and show that this far potential vector can be represented in terms of an equivalent twin lead as shown in Figure 2. By “far” we mean in the transverse x, y direction because this appendix is not limited to finite TL, so for being able to use the results for a semi-infinite TL, we arXiv:1701.04878v3 [physics.class-ph] 25 Jun 2018 keep everything in cylindrical coordinates, and consider the TL between z1 and z2 on the z axis. The twin lead model is defined without loss of generality on the x, z plane and allows Fig. 1. A basic configuration of a two ideal conductors TL, with a well us to define x directed termination currents as current filaments defined separation between the conductors. The surface current distributions [4] across the termination (see Figure 2). Those termination on the contours of the conductors is known from electrostatic considerations, currents contribute a far x directed potential vector, calculated and given that for a two-conductors TL there is only one (differential) TEM mode, the total current is the same on both conductors but with opposite signs. in Appendix A. The arrow shows the vector distance between the center of the surface current In Appendix B we show how to calculate for any cross distributions, named d, obtained for a twin lead equivalent (see Appendix A). section the parameters needed to determine the radiation: the c1,2 are the contours of the “upper” and “lower” conductors, respectively. We consider the case of small electric cross section kd ≪ 1, k is the wavenumber. separation distance d in the twin lead representation, and the characteristic impedance Z0. We perform this analysis on two One of the earliest analysis of radiation from transmission cross section examples. lines (TL) is presented in the paper “Radiation from Trans- It is important to remark that the calculations in this mission Lines” [3], published in 1923. In this publication the work and in [3]–[7] are completely different from what is 2 radiate and rather carries power in the z direction only, but a semi-infinite TL does radiate, and we show in this section the connection between the finite and semi-infinite case and how the transition between them occurs. Here, it is important to mention that in some senses a finite matched TL is very similar to an infinite TL: in both cases there is no reflected wave, but they are very different in what concerns radiation: the first radiates and the second does not. In Section V we discuss the radiation resistance and gener- alize the formula derived in [4]. The work is ended with some concluding remarks. Note: through this work, the phasor amplitudes are RMS Fig. 2. The transmission line is modeled as a twin lead in free space, with values, hence there is no 1/2 in the expressions for power. distance d between the conductors. The currents in the transmission line flow Also, it is worthwhile to mention that the results of this work in the z direction at x = ±d/2 and they contribute to the magnetic vector depend on physical sizes relative to the wavelength, and hence potential Az. The termination currents (source or load) flow in the x direction and contribute the magnetic vector potential Ax. The arrows on the conductors are valid for all frequencies. show the conventional directions of those currents. The wires appear in the figure with finite thickness, but are considered of 0 radius. The transmission line goes in the z direction from z1 to z2. II. POWER RADIATED FROM A FINITE TL A. Matched TL presented as “traveling wave antenna” in many antenna and We calculate in this subsection the power radiated form a electromagnetic books like [9]–[13]. The last ones consider TL of length 2L, carrying a forward wave, represented by the only the current in the “upper” conductor in Figure 2, while current we consider all 4 currents appearing in the figure. This is I(z)= I+e−jkz (1) not an attempt to criticize those works, but only to mention We set z = L and z = L in the expression for their results do not represent radiation from transmission lines, 1 2 the z directed magnetic− vector potential in Eq. (A.13) from hence are not comparable with the results of this work or [3]– Appendix A, and obtain in spherical coordinates: [7]. It should be also mentioned that power loss from TL is Az = µ0G(r)F(z)(θ, ϕ) (2) also affected by nearby objects interfering with the fields, line bends, irregularities, etc. This is certainly true, but those where G is the 3D Green’s function defined in (Eq. (A.2)), affect not only the radiation, but also the basic, “ideal” TL and model in what concerns the characteristic impedance, the F (θ, ϕ)= jI+2Lkd sin θ cos ϕ sinc [kL(1 cos θ)] (3) propagation wave number, etc. Those non-ideal phenomena (z) − are not considered in the current work, nor in [3]–[7], and is the directivity function, and the subscript (z) denotes the also not in [9]–[13]. contribution from the z directed currents. The sinc function The methodology we use for calculating radiation losses is defined sinc(x) sin x/x. To obtain the far fields (those is first order perturbation: we use the lossless (0’th order decaying like 1/r),≡ the ∇ operator is approximated by jkr solution) for the electric current to derive the losses, and and one obtains: − we therefore use in Appendix A the e−jkz dependence. This b H 1 ∇ z methodology is used to derive the ohmic and dielectric losses (z) = (Az )= jkG(r)F(z)(θ, ϕ) sin θϕ (4) µ0 × [10]–[12], [14], and the same approach is used in different b b radiation schemes from free electrons: one uses the 0’th order and E(z) = η0H(z) r, where η0 = µ0/ǫ0 = 120π Ω is the × current (which is unaffected by the radiation) to calculate the free space impedance. For the contributionp of the x directed radiation [17]–[19]. To be mentioned that the same approach end currents we sumb the results for the x directed magnetic has been used in [3]–[7] (although [4] discussed about higher vector potential from Eq. (A.16) in Appendix A (setting z1 = order terms, without applying them). L and z2 = L), obtaining − The main text is organized as follows. In Section II we use A = µ G(r)F (θ, ϕ) (5) the results of Appendix A to calculate the power radiated from x 0 (x) a finite TL carrying a forward wave current, and generalize this where result for any combination of waves.